58edo

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58edo

59edo →

Prime factorization

2 × 29

Step size

20.6897 ¢

Fifth

34\58 (703.448 ¢) (→ 17\29)

Semitones (A1:m2)

6:4 (124.1 ¢ : 82.76 ¢)

Consistency limit

17

Distinct consistency limit

11

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58 equal temperament

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Theory

58edo is a strong system in the 11-, 13- and 17-limit. It is the smallest edo which is consistent through the 17-odd-limit, and is also the smallest distinctly consistent in the 11-odd-limit (the first equal temperament to map the entire 11-odd-limit tonality diamond to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note Genesis scale of Harry Partch.

58et tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539. It supports hemififths, myna, diaschismic, harry, mystery, buzzard, thuja temperaments plus a number of gravity family extensions, and supplies the optimal patent val for the 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments thrush, bluebird, aplonis and jofur.

While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system.

Of all edos which map the syntonic comma (81/80) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1 ¢ narrower than the just interval.

Prime harmonics

Approximation of prime harmonics in 58edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+1.49	+6.79	+3.59	+7.30	+7.75	-1.51	-7.86	-7.58	+4.91	-7.10
Error	Relative (%)	+0.0	+7.2	+32.8	+17.3	+35.3	+37.4	-7.3	-38.0	-36.7	+23.7	-34.3
Steps (reduced)		58 (0)	92 (34)	135 (19)	163 (47)	201 (27)	215 (41)	237 (5)	246 (14)	262 (30)	282 (50)	287 (55)

Subsets and supersets

58 = 2 × 29, and 58edo shares the same excellent fifth with 29edo.

Intervals

#	Cents	Approximate Ratios	Ups and downs notation	Hemipyth notation (SW3-style)
0	0.00	1/1	D	D
1	20.69	56/55, 64/63, 81/80, 128/125	^D, v³E♭
2	41.38	36/35, 49/48, 50/49, 55/54	^^D, vvE♭	α𝄳
3	62.07	26/25, 27/26, 28/27, 33/32	^³D, vE♭
4	82.76	25/24, 21/20, 22/21	vvD♯, E♭
5	103.45	16/15, 17/16, 18/17	vD♯, ^E♭	α
6	124.14	14/13, 15/14, 27/25	D♯, ^^E♭
7	144.83	12/11, 13/12	^D♯, v³E	E𝄳
8	165.52	11/10	^^D♯, vvE
9	186.21	10/9	^³D♯, vE
10	206.90	9/8, 17/15	E	E
11	227.59	8/7	^E, v³F
12	248.28	15/13	^^E, vvF	β𝄳
13	268.97	7/6	^³E, vF
14	289.66	13/11, 20/17	F	F
15	310.34	6/5	^F, v³G♭	β
16	331.03	17/14	^^F, vvG♭
17	351.72	11/9, 16/13	^³F, vG♭	F‡
18	372.41	21/17	vvF♯, G♭
19	393.10	5/4	vF♯, ^G♭	γ
20	413.79	14/11	F♯, ^^G♭
21	434.48	9/7	^F♯, v³G
22	455.17	13/10, 17/13, 22/17	^^F♯, vvG	γ‡
23	475.86	21/16	^³F♯, vG
24	496.55	4/3	G	G
25	517.24	27/20	^G, v³A♭
26	537.93	15/11	^^G, vvA♭
27	558.62	11/8, 18/13	^³G, vA♭	G‡
28	579.31	7/5	vvG♯, A♭
29	600.00	17/12, 24/17	vG♯, ^A♭	δ
30	620.69	10/7	G♯, ^^A♭
31	641.38	13/9, 16/11	^G♯, v³A	A𝄳
32	662.07	22/15	^^G♯, vvA
33	682.76	40/27	^³G♯, vA
34	703.45	3/2	A	A
35	724.14	32/21	^A, v³B♭
36	744.83	20/13, 26/17, 17/11	^^A, vvB♭	ε𝄳
37	765.52	14/9	^³A, vB♭
38	786.21	11/7	vvA♯, B♭
39	806.90	8/5	vA♯, ^B♭	ε
40	827.59	34/21	A♯, ^^B♭
41	848.28	13/8, 18/11	^A♯, v³B	B𝄳
42	868.97	28/17	^^A♯, vvB
43	889.66	5/3	^³A♯, vB	ζ
44	910.34	22/13, 17/10	B	B
45	931.03	12/7	^B, v³C
46	951.72	26/15	^^B, vvC	ζ‡
47	972.41	7/4	^³B, vC
48	993.10	16/9, 30/17	C	C
49	1013.79	9/5	^C, v³D♭
50	1034.48	20/11	^^C, vvD♭
51	1055.17	11/6, 24/13	^³C, vD♭	C‡
52	1075.86	13/7, 28/15	vvC♯, D♭
53	1096.55	15/8, 32/17, 17/9	vC♯, ^D♭	η
54	1117.24	48/25, 40/21, 21/11	C♯, ^^D♭
55	1137.93	25/13, 52/27, 27/14, 64/33	^C♯, v³D
56	1158.62	35/18, 96/49, 49/25, 108/55	^^C♯, vvD	η‡
57	1179.31	55/28, 63/32, 160/81, 125/64	^³C♯, vD
58	1200.00	2/1	D	D

Notation

Sagittal

The following table shows sagittal notation accidentals in one apotome for 58edo.

Steps	0	1	2	3	4	5	6
Symbol

Ups and downs notation

58edo can also be notated using ups and downs notation. In this case, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used to fill in the gaps.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol
Flat symbol

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol
Flat symbol

JI approximation

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 58edo. Prime harmonics are in bold.

As 58edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 58edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/11, 22/13	0.445	2.2
11/10, 20/11	0.513	2.5
15/13, 26/15	0.535	2.6
9/7, 14/9	0.601	2.9
13/10, 20/13	0.958	4.6
15/11, 22/15	0.980	4.7
3/2, 4/3	1.493	7.2
7/6, 12/7	2.095	10.1
9/8, 16/9	2.987	14.4
7/5, 10/7	3.202	15.5
7/4, 8/7	3.588	17.3
11/7, 14/11	3.715	18.0
9/5, 10/9	3.803	18.4
13/7, 14/13	4.160	20.1
11/9, 18/11	4.316	20.9
15/14, 28/15	4.695	22.7
13/9, 18/13	4.762	23.0
5/3, 6/5	5.296	25.6
11/6, 12/11	5.809	28.1
13/12, 24/13	6.255	30.2
5/4, 8/5	6.790	32.8
11/8, 16/11	7.303	35.3
13/8, 16/13	7.748	37.4
15/8, 16/15	8.283	40.0

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3.5	2048/2025, 1594323/1562500	[⟨58 92 135]]	−1.29	1.22	5.89
2.3.5.7	126/125, 1728/1715, 2048/2025	[⟨58 92 135 163]]	−1.29	1.05	5.10
2.3.5.7.11	126/125, 176/175, 243/242, 896/891	[⟨58 92 135 163 201]]	−1.45	1.00	4.83
2.3.5.7.11.13	126/125, 144/143, 176/175, 196/195, 364/363	[⟨58 92 135 163 201 215]]	−1.56	0.94	4.56
2.3.5.7.11.13.17	126/125, 136/135, 144/143, 176/175, 196/195, 364/363	[⟨58 92 135 163 201 215 237]]	−1.28	1.10	5.33

58et has a lower relative error than any previous equal temperaments in the 13-limit, and the next equal temperament that does better in this subgroup is 72.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Period per 8ve	Generator*	Cents*	Associated Ratio*	Temperament
1	3\58	62.07	28/27	Unicorn / alicorn / qilin
1	11\58	227.59	8/7	Gorgik
1	13\58	268.97	7/6	Infraorwell
1	15\58	310.34	6/5	Myna
1	17\58	351.72	49/40	Hemififths
1	19\58	393.10	64/51	Emmthird
1	23\58	475.86	21/16	Buzzard / subfourth
1	27\58	558.62	11/8	Thuja
2	3\58	62.07	28/27	Monocerus
2	1\58	20.69	81/80	Commatic
2	9\58	186.21	10/9	Secant
2	17\58 (12\58)	351.72 (248.28)	11/9 (15/13)	Sruti
2	21\58 (8\58)	434.48 (165.52)	9/7 (11/10)	Echidna
2	24\58 (5\58)	496.55 (103.45)	4/3 (17/16)	Diaschismic
2	25\58 (4\58)	517.24 (82.76)	27/20 (21/20)	Harry
29	19\58 (1\58)	393.10 (20.69)	5/4 (91/90)	Mystery

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

58et can also be detempered to semihemi (58 & 140), supers (58 & 152), condor (58 & 159), and eagle (58 & 212).

Scales

Instruments

Music

Jeff Brown

Fruitbats in Formation

Cam Taylor

58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone

58edo

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